Some Exponential Lower Bounds on Formula-size in Modal Logic

We present two families of exponential lower bounds on the size of modal formulae and use them to establish the following succinctness results. We show that the logic of contingency (ConML) is exponentially more succinct than basic modal logic (ML). We strengthen the known proofs that the so-called public announcement logic (PAL) in a signature containing at least two different diamonds and one propositional symbol is exponentially more succinct than ML by showing that this is already true for signatures that contain only one diamond and one propositional symbol. As a corollary of these results, we obtain an alternative proof of the fact that modal circuits are exponentially more succinct than ML-formulae.